Sufficient conditions for large Galois scaffolds
Let be a finite, Galois, totally ramified -extension of complete local fields with perfect residue fields of characteristic . In this paper, we give conditions, valid for any Galois -group (abelian or not) and for of either possible characteristic ( or ), that are sufficient for the existence of a Galois scaffold. The existence of a Galois scaffold makes it possible to address questions of integral Galois module structure, which is done in a separate paper [BCE]. But since our conditions can be difficult to check, we specialize to elementary abelian extensions and extend the main result of [Eld09] from characteristic to characteristic . This result is then applied, using a result of Bondarko, to the construction of new Hopf orders over the valuation ring that lie in for an elementary abelian -group.
Key words and phrases:Galois module structure, Hopf order
2010 Mathematics Subject Classification:Primary 11S15, Secondary 11R33, 16T05
Let be prime, be a perfect field of characteristic , and be a local field with residue field . Let be a totally ramified Galois extension of with of degree for some , and let be the ring of integers of (i.e. its valuation ring). Local integral Galois module theory asks a question that is a consequence of three classical results: the Normal Basis Theorem, which states that is free over the group algebra ; a result of E. Noether [Noe32], which concludes that, because is wildly ramified, is not free over the group ring ; and a local version of a result of H. W. Leopoldt [Leo59], which states that for absolute abelian extensions of the -adic numbers (i.e. ), is free over its associated order
the largest -order in the group algebra for which is a module.
When is the ring of integers free over its associated order ?
Restrict for the moment to the situation where is a finite extension of . The earliest answers here showed us that unless , need not be free over , which is why the question is currently asked in this way. Additionally, those early answers suggested a form that we might expect the answers to take. Based upon work of F. Bertrandias and M.-J. Ferton [BF72] when is a -extension, and B. Martel [Mar74] when is a -extension, we might expect the answer to Question 1.1, necessary and sufficient conditions for to be free over , to be expressed in terms of the ramification numbers associated with the extension (integers such that where is the th ramification group [Ser79, IV §1]). There have not been that many further results in this direction. Still,
When is an abelian extension, and the ring of integers is replaced with the inverse different , [Byo97, Theorem 3.10] determines necessary conditions, in terms of ramification numbers, for the inverse different to be free over its associated order.
When is unramified and is a totally ramified abelian extension (not necessarily of -power degree), D. Burns [Bur91] investigated freeness of ideals in over their associated orders in . This was extended in [Bur00] to the case where can be ramified, but associated orders are considered in (or, more generally, in , where and is unramified). In both these situations, the existence of any ideal free over its associated order forces strong restrictions on the ramification of the extension .
When is a special type of cyclic Kummer extension, namely for some with , where is the normalized valuation on , Y. Miyata determines necessary and sufficient conditions for to be free over in terms of . These conditions can be restated in terms of ramification numbers [Miy98].
The purpose of this paper is to extend the setting where Galois scaffolds have been proven to exist, namely [Eld09, BE13]: from characteristic to characteristic , and from elementary abelian (or cyclic of degree ) -groups to all -groups (abelian or not). We do this, in Theorem 2.10, by determining conditions sufficient for a Galois scaffold to exist that are independent of characteristic and of Galois group. When an extension satisfies the hypotheses of Theorem 2.10 and thus possesses a Galois scaffold, the answer to Question 1.1 is provided in [BCE], where necessary and sufficient conditions are given, not just for , but for each fractional ideal of , to be free over its associated order. Indeed, stronger questions, such as those asked by B. de Smit and L. Thomas in [dST07], are also addressed. Each answer is given in terms of ramification numbers.
On the other hand, given only the generators of an extension, it is not easy to determine whether the extension satisfies the conditions of Theorem 2.10. Thus in §3, we describe, in terms of Artin-Schreier generators, arbitrarily large elementary abelian -extensions that do satisfy the conditions of Theorem 2.10 and thus possess a Galois scaffold. In characteristic , the result is new. These are the analogs of the near one-dimensional elementary abelian extensions of [Eld09]. In §4, to illustrate the level of explicit detail that is then possible when the results of this paper are combined with [BCE], we include results in characteristic , on the structure of over its associated order, for certain families of elementary abelian extensions that are of common interest.
Finally, to illustrate the utility of our results beyond local integral Galois module theory, we explain how the results of this paper combined with [Bon00, BCE] can be used to attack the difficult problem of classifying Hopf orders in the group algebra for some -group. This is an old problem. The first result in this direction is that of Tate and Oort [TO70] for Hopf orders of rank . And yet, the classifications for remain incomplete [CS05, Proposition 15], [UC06, Theorem 5.4]. Notably, the Hopf orders that are missing for include those which are realizable as the associated orders of valuation rings, and it is precisely such Hopf orders that the results of this paper are designed to produce. Indeed, §5 can be viewed as providing a model, given any -group , for the construction of such “realizable” Hopf orders in . As such, it provides motivation for future work identifying extensions that satisfy the hypotheses of Theorem 2.10.
We close this introduction by pointing out that our work is somewhat similar in spirit to that of Bondarko [Bon00, Bon02, Bon06], who also considers the existence of ideals free over their associated orders in the context of totally ramified Galois extensions of -power degree. Bondarko introduces the class of semistable extensions. Any such extension contains at least one ideal free over its associated order, and all such ideals can be determined from numerical data. Moreover, any abelian extension containing an ideal free over its associated order, and satisfying certain additional assumptions, must be semistable. Abelian semistable extensions can be completely characterized in terms of the Kummer theory of (one-dimensional) formal groups. The precise relationship between Bondarko’s results and our own remains to be explored.
1.1. Discussion of our approach
The existence of a Galois scaffold addresses an issue, which is illustrated in the following two examples. Let denote the normalized valuations for , respectively. Choose with .
Fix a local field and suppose that is a totally ramified Galois extension of degree . Let generate . Then has a unique ramification break , and this is characterized by the property that, for all ,
Let us suppose for simplicity that , say with . Fix a uniformizing parameter of , and let . Pick any with . Then, for , we have . Thus typically reduces valuations by , and the for form an -basis of . Two conclusions follow: firstly, that the form an -basis of the associated order , and, secondly, that is a free module over , generated by any element of valuation .
Example 1.2 in itself is nothing new. Indeed, far more comprehensive treatments of the valuation ring of an extension of degree are given in [BF72, BBF72] for the characteristic case, and in [Aib03, dST07] for characteristic . (See also [Fer73] for arbitrary ideals in characteristic , and [Huy14] and [Mar14] for the corresponding problem in characteristic .) We now consider what happens if we try to make the same argument for a larger extension.
Let be a totally ramified extension of degree . We now have two ramification breaks (in the lower numbering), and we necessarily have . For simplicity we assume that , say , for , . We can then find elements , which generate and for which, setting , we have
whenever . Thus and both typically reduce valuations by , but this does not enable us to determine . Now suppose that we could replace , with elements , such that, for some suitable choice of with , we had
Thus, at least on the family of elements of of the form , we can say that typically reduces valuations by , whilst typically reduces valuations by . We could then deduce that the elements form an -basis of , and that is free over on the generator . Such elements would essentially constitute a Galois scaffold.
The reason that we cannot determine in Example 1.3 using the original elements and is that we have insufficient information about their effect on elements of whose valuation is divisible by but not by . It is because of this problem that early attempts to treat other cases in the same manner as degree extensions achieved only limited success. (See for instance [Fer75] for cyclic extensions of degree , , and, temporarily relaxing the condition that has -power degree, [Fer72, Ber72] for dihedral extensions of degree . A complete treatment of biquadratic extensions of -adic fields was, however, given in [Mar74].)
1.2. Intuition of a scaffold
The intuition underlying a scaffold can be explained, as is done in [BCE], somewhat informally. For the convenience of the reader, we replicate it here: Given any positive integers for such that (think of lower ramification numbers), there are elements such that . Since the valuations, , of the monomials
provide a complete set of residues modulo and is totally ramified of degree , these monomials provide a convenient -basis for . The action of the group ring on is clearly determined by its action on the monomials . So if there were for such that each acts on the monomial basis element of as if it were the differential operator and the were independent variables, namely
then the monomials in the (with exponents bound ) would furnish a convenient basis for whose effect on the would be easy to determine. As a consequence, the determination of , and of the structure of over would be reduced to a purely numerical calculation involving the . This remains true if (2) is loosened to the congruence,
for a sufficiently large “precision” . The , together with the , constitute a Galois scaffold on .
The formal definition of a scaffold [BCE, Definition 2.3] generalizes this situation. Indeed, given this intuitive connection with differentiation, it is perhaps not surprising that scaffolds can be constructed from higher derivations on an inseparable extension, as is done in [BCE, §5]. Ironically, with this perspective it may now be surprising that they can be constructed for Galois extensions under the action of . Yet, this is where they were first constructed [Eld09].
2. Main Result: Construction of Galois scaffold
Recall that is a complete local field whose residue field is perfect of characteristic , and that is a totally ramified Galois extension of degree . Relabel now, so that . Following common practice, we use subscripts to denote field of reference. So is the normalized valuation, and is a prime element of with . The valuation ring of is denoted by with maximal ideal . Let be the th group in the ramification filtration of the Galois group .
In this section we construct a Galois scaffold, in Theorem 2.10, for extensions that satisfy three assumptions, which in turn depend upon two choices. For emphasis, we repeat here that may have characteristic or . The Galois group can be nonabelian, as well as abelian. We also point out that, except for Assumption 2.9, all these choices and assumptions appear in [Eld09]. Our first choice organizes the extension.
Choose a composition series for that refines the ramification filtration: such that , and . Furthermore, choose one element to represent each degree quotient: .
Let be the fixed field of , and let . Because of Choice 2.1, we can see, using [Ser79, IV§1], that the multiset is the set of lower ramification numbers, namely the set of subscripts with , with multiplicity . In particular, , is the ramification multiset for , is the ramification multiset for , and is the lower ramification number for . The set of upper ramification numbers is determined by
[Ser79, IV§3]. Furthermore note that is the set of upper ramification numbers for , but that is not necessarily the set of upper ramification numbers for .
Our first assumption is weak, as it does not eliminate any extension in characteristic . In characteristic , it eliminates only maximally ramified extensions, i.e. those cyclic extensions where contains the th roots of unity and for some prime element [Ser79, IV§2 Exercise 3].
Now we choose generators for based upon Choice 2.1. Since the valuation is normalized so that , there are with . Since , a unit exists such that .
For each , choose such that and .
Since [Ser79, IV§2], we have and therefore .
Since , we could choose so that, additionally, it satisfies an Artin-Schreier equation [FV02, III §2 Proposition 2.4]. In characteristic , this is a result of MacKenzie and Whaples. We do not make this a requirement however, since we do not need to use this fact.
Define the binomial coefficient
for , and for . For integers form the -tuple, . Define . Thus , if there is an with . Define the partial order on -tuples: Given , ,
Thus if and only if . Now restrict to vectors of the base- coefficients of integers , and identify each where with its corresponding vector. (It is convenient to index the base- digits as , where increasing values of correspond to decreasing powers of .) Define
Because the are relatively prime to , is a complete set of residues modulo . As a result, is a -basis for . Since maps the residues modulo onto the residues modulo , it has an inverse : For each , we define to be the unique integer satisfying
Note that . Using this notation, we normalize our -basis for as follows.
Let , where is a fixed prime element in . Thus for all , , and is an -basis for .
with . We consider to be the main term, with the error term. Observe that . We would like this observation to be a statement about an element that approximates the effect of . So observe that if , then we may choose . The condition for all is equivalent to
There is one residue class modulo , represented by with , such that for .
Under this assumption . Furthermore, . Restated in terms of upper ramification numbers, Assumption 2.6 becomes for . Since , this implies the conclusion of the Theorem of Hasse-Arf, namely that the upper ramification numbers are integers. But Assumption 2.6 is stronger than the conclusion of Hasse-Arf, since it implies that the upper ramification numbers are integers congruent modulo .
Define truncated exponentiation by
where is the integers localized at . Motivated by [Eld09], we define:
Let where , and for ,
where the usual vector space operations of addition and scalar multiplication have been replaced by multiplication and scalar truncated exponentiation, respectively. Note for all in the augmentation ideal . So, since satisfies , we find . A cautionary remark is important here: Since scalar truncated exponentiation does not distribute (it is easy to check for that the units and are not equal), applying the inverse matrix to both sides of this equation does not preserve equality.
The following assumption will enable us to ignore the error terms in (5).
Given an integer , assume that for ,
which because of (4), is equivalent to .
We state the main result of this paper.
Let be a totally ramified Galois -extension with ramification multiset satisfying Assumptions 2.2 and 2.6. Thus there is one congruence class modulo , represented by , that contains all the ramification numbers. Given an integer , assume that it is possible to make Choices 2.1 and 2.3 so that Assumption 2.9 holds. Then a -scaffold on , as defined in [BCE, Definition 2.3], exists with precision and shift parameters . Namely, there are:
Before we prove this theorem, some discussion of the elements is warranted. Suppose satisfies where is the element of that gives the trace from to . In this case, we will say that is a lift of . Thus can be considered to be one among many lifts of . Now observe that for with and thus . The following result states that is a natural upper bound on for a lift of . From this perspective, Theorem 2.10 states that the lifts , provided by Definition 2.7, are special in that they achieve a natural upper bound.
The case is trivial since we necessarily have . Fix and consider the different of the extension . Hilbert’s formula for the exponent of the different [Ser79, IV§1 Proposition 4] gives where . For any , we have where and denotes the greatest integer . Since by Assumption 2.6, it follows that
In particular, if for some , we find that and . Let with . We may write an arbitrary element as with and . Since , it follows that , and hence that provided that . Recalling Assumption 2.2, we have therefore shown that if but then .
Now, with as above, suppose that is any element of with . Since , we have , so that . Hence . ∎
We conclude this section by recording a technical question.
We do not address this question here. Neither was it addressed in [Eld09, BE13]. Thus far, in all these cases, naive choices were made that turned out to be good enough for a determination of Galois module structure. There has been no need.
2.1. Proof of Theorem 2.10
for with . Our analysis is technical. To motivate it, we begin by considering the special case treated in [Eld09] where for all . This gives us the opportunity to more fully justify [Eld09, (4)]. Observe that Theorem 2.10 with follows from Proposition 2.13 setting .
Note that fixes for . So it is sufficient to prove by inducting down from to that . Recall . Pascal’s Identity states that . Thus for . Recall . Observe that
where the last equality is a consequence of Vandermonde’s Convolution Identity, . Thus because we have
So , based upon Pascal’s Identity. Note the role of Pascal’s Identity and Vandermonde’s Convolution Identity. These two identities will be used repeatedly and without mention in the induction step.
Assume that the proposition holds for all such that . Thus for each with ,